# Solving simultaneous equations by various methods

Write a system of two equations in two unknowns for each problem.

Solve each system by substitution. See Examples 8 and 9.

92. Investing her bonus. Donna invested her $33,000 bonus and received a total of $970 in interest after one year. If part of the money returned 4% and the remainder 2.25%, then how much did she invest at each rate?

100. Ticket sales. Tickets for a concert were sold to adults for $3 and to students for $2. If the total receipts were $824 and twice as many adult tickets as student tickets were sold, then how many of each were sold?

104. Bonus and taxes. A company has an income of $100,000 before paying taxes and a bonus. The bonus B is to be 20% of the income after deducting income taxes T but before deducting the bonus. So B = 0.20(100,000 - T ).

Because the bonus is a deductible expense, the amount of income tax T at a 40% rate is 40% of the income after deducting the bonus. So T = 0.40(100,000 - B).

a) Use the accompanying graph to estimate the values of T and B that satisfy both equations.

b) Solve the system algebraically to find the bonus and the amount of tax.

106. Free market. The equations S = 5000 + 200x and D = 9500 - 100x express the supply S and the demand D, respectively, for a popular compact disc brand in terms of its price x (in dollars).

a) Graph the equations on the same coordinate system.

b) What happens to the supply as the price increases?

c) What happens to the demand as the price increases?

d) The price at which supply and demand are equal is called the equilibrium price. What is the equilibrium price?

Solve each system by substitution or addition, whichever is easier.

Write a system of two equations in two unknowns for each problem. Solve each system by the method of your choice.

72. Books and magazines. At Gwenâ??s garage sale, all books were one price, and all magazines were another price. Harriet bought four books and three magazines for $1.45, and June bought two books and five magazines for $1.25. What was the price of a book and what was the price of a magazine?

78. Low-fat yogurt. Ziggyâ??s Famous Yogurt blends regular yogurt that is 3% fat with its no-fat yogurt to obtain lowfat yogurt that is 1% fat. How many pounds of regular yogurt and how many pounds of no-fat yogurt should be mixed to obtain 60 pounds of low-fat yogurt?

82. Super Bowl contender. The probability that San Francisco plays in the next Super Bowl is nine times the probability that they do not play in the next Super Bowl. The probability that San Francisco plays in the next Super Bowl plus the probability that they do not play is 1. What is the probability that San Francisco plays in the next Super Bowl?

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#### Solution Preview

92>

Let the bonus money be split into two portions: X invested at a return of 4% and Y invested at a return of 2.25%

We are told that the total bonus is $33,000 so we can first of all say that the portions X and Y added together amount to this

bonus amount

So

X + Y = 33,000 (1)

Using the fact that the total interest received in one year is $970 we can say that 4% of the invested amount X plus 2.25%

of the invested amount Y = $970 So a second equation describing this would be (2)

{4/100}*X + {2.25/100}*Y = 970 (2)

We have 2 equations and 2 unknowns so we can proceed to solve for the unknowns X and Y

First lets us get rid of the 100 denominator from LHS of (2). To do this we must multiply both LHS and RHS of (2) by 100.

We then get (3)

4X + 2.25Y = 97000 (3)

Now on to the substitution method to solve for X, Y

Rearrange (1) to give X in terms of Y. We have done this in (4)

X = 33,000 - Y (4)

Now we substitute for everywhere we see X (as given by (4)) in equation (3)

4*(33000 - Y) + 2.25Y = 97000 (5)

Multiplying out the bracket

132000 - 4Y + 2.25Y = 97000

132000 - 1.75Y = 97000

1.75Y = 132000 - 97000

1.75Y = 35000

Y = 35000/1.75

Y = $20,000

So if Y = $20,000 we resubstitute for this in (4) to get

X = 33,000 - 20,000 = $13,000

100>

Let the total number of Adult tickets sold be represented by A, Total number of Students tickets be represented by S

the total receipts were $824

As tickets for Adults cost $3 each monies received for sale of A adult tickets must be A x $3 or 3A (omitting the dollar signs)

Similarly we are told that the tickets for Students cost $2 each so total monies received for sale of S Students tickets must

be S x $2 or 2S (omitting the dollar signs)

Therefore we can ...

#### Solution Summary

A selection of various word descriptions, formation of the relevant simultaneous equations and solving the simultaneous equations using different methods are shown